A parabolic almost monotonicity formula

We prove the parabolic counterpart of the almost monotonicity formula of Caffarelli, Jerison and Kening for pairs of functions u _±(x, s) in the strip satisfying

Finite groups in which all p-subnormal subgroups form a lattice

O. Kegel, in 1962, introduced the concept of p-subnormal subgroups of a finite group as the subgroups whose intersections with all Sylow p-subgroups of the group are Sylow p-subgroups of the subgroup. The set of p-subnormal subgroup of a finite group is not a lattice in general. In this paper, the class of all finite groups in which all p-subnormal subgroups from a lattice is determined. This is the class of all finite p-soluble groups whose p-length and p′-length, both, are less or equal to 1. The join-semilattice case and the meet-semilattice case are analyzed separately. The authors are supported by Proyecto PB 94-1048 of DGICYT, Ministerio de Educación y Ciencia of Spain.     

On capability of finite abelian groups

Baer characterized capable finite abelian groups (a group is capable if it is isomorphic to the group of inner automorphisms of some group) by a condition on the size of the factors in the invariant factor decomposition (the group must be noncyclic and the top two invariant factors must be equal). We provide a different characterization, given in terms of a condition on the lattice of subgroups. Namely, a finite abelian group G is capable if and only if there exists a family {H _i } of subgroups of G with trivial intersection, such that the union generates G and all quotients G/H _i have the same exponent. Other variations of this condition are also provided (for instance, the condition that the union generates G can be replaced by the condition that it is equal to G). The work presented here is partially supported by NSF/DMS-0805932.     

Automorphisms fixing elements of prime order in finite groups

Let σ be an automorphism of a finite group G and suppose that σ fixes every element of G that has prime order or order 4. The main result of this paper shows that the structure of the subgroup H=[G, σ] is severely limited in terms of the order n of σ. In particular, H has exponent dividing n and it is nilpotent of class bounded in terms of n.     

Discontinuous extensions and almost resolvability

It is proven that in large classes of topological spaces each real-valued continuous function on aG _δ-setG has an extension to the whole space which is continuous exactly at the points ofG. AmongG _δ-spaces this property characterizes the almost resolvable normal spaces.     

Groups with complemented nonmodular subgroups

We give the structure of finite groups belonging to some families defined by conditions of complementation. The author was supported by Istituto Nazionale di Alta Matematica “Francesco Severi” - Rome.     

On the almost sure central limit theorem and domains of attraction

We give necessary and sufficient criteria for a sequence (X _n) of i.i.d. r.v.'s to satisfy the a.s. central limit theorem, i.e.,

Recurrent random walks on homogeneous spaces of p-adic algebraic groups of polynomial growth

Let G be a p-adic algebraic group of polynomial growth and H be a closed subgroup of G. We prove the growth conjecture for the homogeneous space G/H, that is, G/H supports a recurrent random walk if and only if G/H has polynomial growth of degree atmost two. Received: 23 November 2007     

Localization of blow-up points for a parabolic system with a nonlinear boundary condition

The authors localize the blow-up points of positive solutions of the systemu _t =Δu,v _t =Δv with conditions at the boundary of a bounded smooth domain Θ under some restrictions off andg and the initial data (Δu ₀, Δν₀>c>0). If Θ is a ball, the hypothesis on the initial data can be removed. Supported by Universidad de Buenos Aires under grant EX071 and CONICET.     

Characterisations of A p+2 and L 2(2 q )

Let G be a transitive group of degree p+2 with p|G| where p ≧ 5 is a prime number, then (i) G is isomorphic to S _p+2 or A _p+2, if G has an element of order 4, (ii) G is isomorphic to L ₂(2 ^q ) or P Γ L ₂ (2 ^q ), if 2 ^q − 1=p is a Mersenne prime and G has no element of order 4.     

Semifree actions of free groups

We study countable universes similar to a free action of a group G. It turns out that this is equivalent to the study of free semi-actions of G, with two universes being transformable iff one corresponding free semi-action can be obtained from the other by a finite alteration. In the case of a free group G (in finitely many or countably many generators), a classification is given. Research partially supported by a DAAD Doktorandenstipendium D/02/02345.     

Note on hit-and-miss topologies

This is a continuation of [19]. We characterize first and second countability of the general hit-and-miss hyperspace topologyτ ₊ ^Δ for weakly-R ₀ base spaces. Further, metrizability ofτ ₊ ^Δ is characterized with no preliminary conditions on the base space and the generating family of closed sets and a new proof on uniformizability (i.e. complete regularity) ofτ ₊ ^Δ is given in this general setting, thus generalizing results of [3], [5] and [6].     

Finite groups with minimal 1-PIM

Let be a field of characteristic and let G be a finite group. It is well-known that the dimension of the minimal projective cover (the so-called 1-PIM) of the trivial left -module is a multiple of the -part of the order of G. In this note we study finite groups G satisfying . In particular, we classify the non-abelian finite simple groups G and primes satisfying this identity (Theorem A). As a consequence we show that finite soluble groups are precisely those finite groups which satisfy this identity for all prime numbers (Corollary B). Another consequence is the fact that the validity of this identity for a finite group G and for a small prime number implies the existence of an -Hall subgroup for G (Theorem C). An important tool in our proofs is the super-multiplicativity of the dimension of the 1-PIM over short exact sequences (Proposition 2.2).     

Existence theorems for nonlinear functional differential equations of neutral type

Conditions are found upon satisfaction of which the differential equation

Algebraic extensions of semi-hyponormal operators

In this paper, we show that algebraic extensions of semi-hyponormal operators (defined below) are subscalar. As corollaries we get the following:

Automorphism groups of semidirect products

This paper shows that if is a semidirect product of finite groups, then if and only if and for all . As an application, we investigate the automorphism group of a split metacyclic p-group for odd p. The second author is supported by the Natural Science Foundation of China (10671058).     

The parity problem of polymatroids without double circuits

According to the present state of the theory of the matroid parity problem, the existence of a good characterization to the size of a maximum matching depends on the behavior of certain substructures, called double circuits. In this paper we prove that if a polymatroid has no double circuits then a partition type min-max formula characterizes the size of a maximum matching. Applications to parity constrained orientations and to a rigidity problem are given. Research is supported by OTKA grants K60802, TS049788 and by European MCRTN Adonet, Contract Grant No. 504438.     

Advertible complétude et structure deQ-algèbre

We give different characterizations of algebra types (Q-property, advertible completeness, MackeyQ-property) in terms of the relation ρ<-β and sequentiality notions in locally pseudo-convex algebras.

     

Free products of finitary skew linear groups

We give two very different proofs of the following result. Let {G _γ:λɛΛ : λ ∈ Λ} be a family of finitary skew linear groups of the same characteristic p ≧ 0. Then the free product of the G _γ is isomorphic to some finitary skew linear group of characteristic p. This extends recent work of R. J. H. Minty on the skew linear case and of O. Puglisi on the finitary linear case.